{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T03:52:18Z","timestamp":1761709938542,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T00:00:00Z","timestamp":1655769600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Hebei Province","award":["A2020402006","11701136","12001156"],"award-info":[{"award-number":["A2020402006","11701136","12001156"]}]},{"name":"National Natural Science Foundation of China","award":["A2020402006","11701136","12001156"],"award-info":[{"award-number":["A2020402006","11701136","12001156"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar graphs. Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any two adjacent vertices admit different sets consisting of the colors of their incident edges. The neighbor-distinguishing index \u03c7a\u2032(G) of G is the smallest number of colors in such an edge coloring of G. It was conjectured that if G is a connected graph with at least three vertices and G\u2260C5, then \u03c7a\u2032(G)\u2264\u0394+2. In this paper, we show that if G is a planar graph with maximum degree \u0394\u226513, then \u0394\u2264\u03c7a\u2032(G)\u2264\u0394+1, and, further, \u03c7a\u2032(G)=\u0394+1 if and only if G contains two adjacent vertices of maximum degree.<\/jats:p>","DOI":"10.3390\/sym14071289","type":"journal-article","created":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T23:11:19Z","timestamp":1655939479000},"page":"1289","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Characterization for the Neighbor-Distinguishing Index of Planar Graphs"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3299-6060","authenticated-orcid":false,"given":"Jingjing","family":"Huo","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hebei University of Engineering, Handan 056038, China"}]},{"given":"Mingchao","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hebei University of Engineering, Handan 056038, China"}]},{"given":"Ying","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1016\/S0893-9659(02)80015-5","article-title":"Adjacent strong edge coloring of graphs","volume":"15","author":"Zhang","year":"2002","journal-title":"Appl. Math. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1137\/S0895480102414107","article-title":"Adjacent vertex distinguishing edge-colorings","volume":"21","author":"Balister","year":"2007","journal-title":"SIAM J. Discrete Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3005","DOI":"10.1016\/j.disc.2004.12.027","article-title":"r-Strong edge colorings of graphs","volume":"306","author":"Akbari","year":"2006","journal-title":"Discrete Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.dam.2013.08.038","article-title":"An improved upper bound on the adjacent vertex distinguishing chromatic index of a graph","volume":"162","author":"Zhang","year":"2014","journal-title":"Discrete Appl. 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Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2412","DOI":"10.1137\/120903178","article-title":"A characterization on the adjacent vertex distinguishing index of planar graphs with large maximum degree","volume":"29","author":"Wang","year":"2015","journal-title":"SIAM J. Discrete Math."},{"key":"ref_12","first-page":"313","article-title":"Adjacent vertex-distinguishing edge coloring of graphs","volume":"16","author":"Bonamy","year":"2013","journal-title":"Seventh Eur. Conf. Combin. Graph Theory Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1007\/s40840-021-01213-9","article-title":"On the neighbor-distinguishing indices of planar graphs","volume":"45","author":"Wang","year":"2022","journal-title":"Bull. Malays. Math. Sci. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1289\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:36:52Z","timestamp":1760139412000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1289"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,21]]},"references-count":13,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071289"],"URL":"https:\/\/doi.org\/10.3390\/sym14071289","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,6,21]]}}}